1900 Progress of Theoretical Physics, Vol. 57, No. 6, June 1977 Theory of Canonical Transformations for Nonlinear Evolution Equations. II Yuji KODAMA Department of Physics, Nagoya University, Nagoya 464 (Received November 29, 1976) A canonical transformation for nonlinear evolution equations is discussed in relation to the inverse scattering scheme. It is shown that completely integrable Hamiltonian systems have the canonical transformation of the type, which keeps the forms of the conserved quantities of those systems invariant, and that the Backlund transformations associated with those systems are canonical transformations. § 1. Introduction In a previous paper, 1l the author and Wadati investigated the canonical trans- formation for the nonlinear evolution equation of the form (1·1) Equation (1·1) contains the sine-Gordon equation, the Korteweg-de Vries equation and the modified Korteweg-de Vries equation as special cases. The canonical trans formation considered there keeps the form of the Hamiltonian invariant, that is, the transformed Hamiltonian is formally identical with the original Hamiltonian except for an additional constant. It was shown that the Backlund transformations associated with the nonlinear equations of the form (1·1) are the canonical trans- formations of that type, and that the infinite number of conservation laws are derived in terms of the infinitesimal canonical transformations. On the other hand, it is well known that the nonlinear evolution equation, which can be solved by means of the inverse scattering method, possesses the infinite number of conservation laws.") Indeed, those conservation laws are derived by using the inverse scattering scheme. 3 l Thus, the existence of the infinite con- servation laws is closely connected with the applicability of the inverse scattering problem. We note that each of those nonlinear evolution equations is expressed in terms of the Hamiltonian system with appropriate one of those conserved quan- tities as the Hamiltonian. 4 l In relation to this fact, recently, several authors have reported that those nonlinear equations are completely integrable. 5 l These results suggest that the applicability of the inverse scattering method leads very naturally to the complete integrability of the system, and that it is closely connected with the existence of the canonical transformation 6 l through the existence of the infinite number of conservation laws. Downloaded from https://academic.oup.com/ptp/article/57/6/1900/1939173 by guest on 13 June 2022